open problems survey

[Catullus]

Conjecture:

The standard extension of exponentiation is the only extension of exponentiation, such that a to the power of x is totally monotonic, For all real a between one and zero non inclusive, and for all real a.

OHHHHHH CATULLUS!!!

This is a good one.

I'd suggest an old thread by me on MathOverflow, but it's lost in all the overflow.

I think that this is the thread.

[Catullus]

Conjecture:

\(\forall a,b\in\Bbb N-1\:\lim_{t\to\infty}g_a(b\uparrow\uparrow\lfloor t\rfloor)\in\overline{\Bbb Q}\iff b\div a\in\Bbb N\), where \(g_a\) denotes the a-ary van der Corput sequence.

[leon]

Conjecture:

\(\forall a,b\in\Bbb N-1\:\lim_{t\to\infty}g_a(b\uparrow\uparrow\lfloor t\rfloor)\in\overline{\Bbb Q}\iff b\div a\in\Bbb N\), where \(g_a\) denotes the a-ary van der Corput sequence.

the matrix for the logic sequence that outputs the variable x > pentation

[leon]

TPID 18: Is "polynomial" tetration (by truncated Carleman-matrices) of some order \(n\) with \( n \to \infty \) asymptotic to the Kneser-solution? 

This is a long outstanding problem of mine. I've some numerical examples since years which seem to suggest that, but had never an idea how to approach at all a proof/disproof of such a conjecture.          

One example where I worked with this is here but I've some more material of comparision in my local excel- and Pari/gp-files which might be worked out further.         
My examples with truncations of the Carlemanmatrices from sizes \(4 \times 4,8 \times 8,16 \times 16,32 \times 32 \) up to \(64 \times 64 \) with some random chosen small bases (like \(b=\sqrt 2 , b=4 , b=e \) ) show nice improvement of approximations to 5 or 10 ore more correct digits.

sorry to ask here , but has it been proven that all rising integer sequences (of size) give the same result ?

regards

tommy1729

it has not been proven yet because when people try with riemannian and gaussian methods they come up short